tr(a + B) = tr(a) + tr(b) tr(ca) = c tr(a)

Transcription

1 Chapter 3 Determinant 31 The Determinant Funtion We follow an intuitive approah to introue the efinition of eterminant We alreay have a funtion efine on ertain matries: the trae The trae assigns a numer to a square matrix y summing the entries along the mail iagonal of the matrix So the trae is a funtion; its omain is the set of all square matries, its range is the set of numers We also showe that the trae is a linear funtion, that is tra + B = tra + trb tra = tra where A an B are two n n matries, an is a onstant The eterminant is also a funtion whih assigns a numer to every square matrix So its omain will e again the set of square matries, an its rage is the set of numers The notation of the eterminant of a matrix A is eta The eterminant funtion has some nie properties, an we shoul emphasize two of them at this point: etab = eta etb eti = 1 where A an B are two n n matries We will see the other properties later in this hapter

2 38 Determinant 32 Calulating the Determinant for 2 2 Matries Definition 321 The eterminant of a 2 2 matrix is efine as a et = a Example 321 Example eti 2 = et = et = = Theorem 321 For any two 2 2 matries A an B etab = eta etb Proof Let a 11 a A = an B = a 21 a Then eta etb = a 11 a 22 a 12 a = a 11 a a 11 a a 12 a a 12 a The prout of the two matries is a a a a AB =, a a a a so etab = a a a a a a a a = a a a a a a a a a a a a a a a a = a a a a a a a a 21 11, 322

3 32 Calulating the Determinant for 2 2 Matries 39 As we see the terms in 321 an in 322 are the same, so eta etb = etab If A is an invertile matrix, then we an apply this theorem for A an A 1 : etaa 1 = eta eta 1 The left han sie of the equation is etaa 1 = eti = 1, so we have that 1 = eta eta 1 From this equation we an onlue that if A is invertile, then eta 0, eta 1 0, an eta 1 = Corollary 322 If A is invertile, then eta 1 = 1 eta 1 eta Using the efinition of the eterminant we an easily show some properties of the eterminant funtion for 2 2 matries Corollary et = Corollary 324 If a row or olumn of a matrix is 0, then the eterminant of the matrix is 0 Corollary 325 If a matrix is triangular, then its eterminant is the prout of the iagonal entries: a 0 et = a, a et = a 0

4 40 Determinant Corollary 326 If we swap two rows or two olumns, then the eterminant hanges its sign: et a et a = et = et a a, Corollary 327 If we multiply a row or olumn y a numer k, then the eterminant will e k times as efore: et et ka k = k et = k et k ka a a, Corollary 328 If A is a 2 2 matrix an is a salar, then eta = 2 eta Corollary 329 If we a k times a row or olumn to the other row or olumn, then the eterminant will no hange: a + k + k et = a + k + k = a a = et 33 Geometri Meaning of the Determinant We an onsier a 2 2 matrix x 1 y 1 as a olletion of two olumn vetors x = x 1 x 2 x 2 y 2 an y = x 1 x 2

5 34 Properties of the Determinant Funtion 41 1,08 2,08 0,0 1,0 Figure 31: Area of a parallelogram We an raw these two vetors in the Desartes oorinate system, an we an see that any two suh vetors etermine a parallelogram You may see Setion 41 for more aout vetors Example 331 The area of the parallelogram with verties 0, 0, 1, 0, 1, 08 an 2, 08 is 08, see Figure 31 This is the same as the eterminant of the matrix forme y the two olumn vetors whih etermine the parallelogram: 1 1 et = Example 332 The area of the parallelogram with verties 1, 1, 2, 1, 2, 18 an 3, 18 is the same as in the previous example, sine this parallelogram is just shifte, ut its area has not hange x 1 y 1 Theorem 331 The eterminant of a 2 2 matrix is the signe area of x 2 y 2 the parallelogram etermine y the two olumn vetors of the matrix The sign is positive if the angle that you get y rotating the first olumn vetor towar the seon olumn vetor in the ounterlokwise iretion is less than π The sign is negative if this angle is greater than π Proof We will only give a visual verifiation of this theorem here y utting an moving piees of the parallelogram aroun, see Figure 32 On eah piture, the ark shae part has the same area as the lightly shae piee

6 42 Determinant x1+x2,y1+y2 y2 y1,y2 x2 x1,x2 y1 x1 a y2 x2 y2 x2 y1 x1 y1 x1 y2 A B y2 x2 A B x2 O y1 x1 y1 x1 e Figure 32: Verifying Theorem 331

7 34 Properties of the Determinant Funtion Properties of the Determinant Funtion We efine the eterminant funtion for 2 2 matries, an saw some nie properties of it We woul like to exten our efinition for all square matries, so that these properties remain true Theorem 341 The eterminant assigns a numer for every square matrix, with the following properties Let A an B e two n n matries 1 eti n = 1 2 If B is the matrix that results when a single row or single olumn of A is multiplie y a salar k, then etb = k eta 3 If B is the matrix that results when two rows or two olumns of A are interhange, then etb = eta 4 If B is the matrix that results when a multiple of one row is ae to another row or when a multiple of one olumn of A is ae to another olumn, then etb = eta Some further properties of the eterminant that we showe for 2 2 matries an remain true for larger matries: Theorem 342 Let A an B e n n matries 1 If A has a row or a olumn of zeroes, then eta = 0 2 If A is a triangular matrix, then its eterminant is the prout of the iagonal entries 3 eta = eta T 4 etab = eta etb 5 A square matrix A is invertile if an only if eta 0 6 If A is invertile, then eta 1 = 1 eta 7 If k is a salar, then etka = k n eta

8 44 Determinant 35 Evaluating the Determinant y Row Reution Theorem 341 allows us to use row- or olumn- reution to alulate the value of the eterminant of larger matries than 2 2 The goal is to reue the matrix to a triangular form, eause we know that the eterminant of a triangular matrix is the prout of its entries along the main iagonal Example 351 Using row reution let s alulate the eterminant of the matrix x x 2 x x 3 x 3 x 3 1 If we o the following row operations R 2 = R 2 xr 1, R 3 = R 3 x 2 R 1, R 4 = R 4 x 3 R 1, then the value of the eterminant oes not hange x et x 2 x = et 0 1 x x 2 1 x 3 x 3 x x 3 = 1 x1 x 2 1 x 3 Example 352 Sine we alreay alulate the eterminant of x A = x 2 x 2 1 1, x 3 x 3 x 3 1 we an easily tell that this matrix is singular has no inverse over the real numers if the eterminant is equal to zero, that is if x = ±1 Notie, that eta = 0 has two more omplex roots, so over the omplex numers A is singular not only for x = ±1 ut also for x = e i 2π 3, or x = e i 4π 3

9 36 Determinant, Invertiility an Systems of Linear Equations Determinant, Invertiility an Systems of Linear Equations Every system of linear equations an e written in a matrix form: A x =, where A is the oeffiient matrix, x is the vetor of unknowns, an is the vetor of onstant terms You an always solve a system of linear equations using the Gaussian algorithm, y reuing it to row ehelon form or to the reue row ehelon form However if the oeffiient matrix A is a square matrix, then alulating the eterminant might e useful Theorem 361 If A is an n n matrix, then the following are equivalent That is, if one of these statements is true, then all the others must also e true 1 A is invertile 2 eta 0 3 A x = has exatly one solution for every n 1 matrix 4 A x = 0 has only the trivial solution that is the solution is the zero vetor 5 The reue row-ehelon form of A is I n Proof The iea of the proof: if A is invertile, then we an multiply oth sies of the equation A x = y A 1, an we get A 1 A x = A 1 Sine A 1 A = I, we solve the equation: x = A 1, an there is only one solution for every If was the zero vetor, then the solution is x = A 1 0 = 0, the trivial solution eah of its omponent is 0

10 46 Determinant Corollary 362 If A is an n n matrix suh that eta = 0, then the equation 1 A x = 0 has a non-trivial solution that is a solution whose omponents are not all 0 2 A x = has either more than one solution for a non-zero n 1 matrix, or has no solutions at all We have to use the Gaussian algorithm to fin out what is happening in this ase Example 361 The linear system 2kx + k + 1y = 2 k + 6x + k + 3y = 3 has exatly one solution if 2k k + 1 et 0, k + 6 k + 3 that is when 2kk + 3 k + 1k + 6 = k 2 k 6 = k 3k + 2 0, ie if k 2, 3 However if k = 2 or 3, then we have to use the Gaussian algorithm If k = 2, then the system eomes 4x y = 2 4x + y = 3 whose row ehelon form is 1 1/4 1/2, an we an onlue that the system has no solution If k = 3, then the system eomes 6x + 4y = 2 9x + 6y = 3 whose row ehelon form is 1 2/3 1/ In this ase y is a free variale, an the system has infinitely many solutions over R an also over C

11 37 Cofator Expansion, Ajoint Matrix Cofator Expansion, Ajoint Matrix Definition 371 The reursive efinition of the eterminant using ofator expansion along the ith row of A: et A = a i1 C i1 + a i2 C i2 + a i3 C i3 + a in C in With sum notation: n eta = 1 i+k a ik C ik k=1 The reursive efinition of the eterminant using ofator expansion along the jth olumn of A: et A = a 1j C 1j + a 2j C 2j + a 3j C 3j + a nj C nj With sum notation: n eta = 1 k+j a kj C kj k=1 Here C ij enotes the ofator of the entry a ij : the eterminant of the minor you get from A y anelling the ith row an jth olumn, with a plus or minus sign aoring to the hekeroar : Definition 372 The ofator matrix of A: C 11 C 12 C 13 C 1n C 21 C 22 C 23 C 2n C 31 C 32 C 33 C 3n C n1 C n2 C n3 C nn is the matrix you get y replaing eah entry in A y its ofator

12 48 Determinant Definition 373 The ajoint matrix of A is the transpose of the ofator matrix: C 11 C 21 C 31 C n1 C 12 C 22 C 32 C n2 aja = C 13 C 23 C 33 C n3 C 1n C 2n C 3n C nn Theorem 371 eta eta 0 0 A aja = = eta I n eta Theorem 372 If A is an invertile matrix, then Example 371 Let A 1 = 1 eta aja A = To get the eterminant of A we an use ofator expansion The first row woul e the est hoie, sine it has a zero in it an the other entries are 1, whih makes the alulations easier: eta = a 11 C 11 + a 12 C 12 + a 13 C = 1 et et = = 6 The ofator matrix of A is:

13 38 Calulating the Determinant for 3 3 Matries 49 The ajoint of A is: aja = Calulating A aja: A aja = = = eta I Therefore we an use the ajoint matrix to fin the inverse of A: A 1 1 = eta aja = /6 5/ = 2 1/6 5/ /6 1/6 38 Calulating the Determinant for 3 3 Matries Theorem 381 The Sarrus s rule To fin the eterminant of a 3 3 matrix A, write the first two olumn of A to the right of A Then, multiply the entries along as the iagram shows, then a or sutrat these prouts: a 11 a 12 a 13 a 11 a 12 a 21 a 22 a 23 a 21 a 22 a 31 a 32 a 33 a 31 a 32 eta = a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 a 13 a 22 a 31 a 11 a 23 a 31 a 12 a 21 a 33 Please note, that Sarrus s rule works only for 3 3 matries For larger matries you either have to use row- or olumn reution, ofator expansion or see if the matrix is lok triangular

14 50 Determinant 39 Blok-Triangular Matries If A is a lok-triangular matrix, then the eterminant of A is the prout of the eterminants of its iagonal loks Example et = et et 1 et = = 6